Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings

Ce Chidume,Naseer Shahzad,Habtu Zegeye

doi:10.1155/fpta.2005.233

Abstract

Let be a nonempty closed convex subset of a reflexive real Banach space which has a uniformly Gâteaux differentiable norm. Assume that is a sunny nonexpansive retract of with as the sunny nonexpansive retraction. Let , , be a family of nonexpansive mappings which are weakly inward. Assume that every nonempty closed bounded convex subset of has the fixed point property for nonexpansive mappings. A strong convergence theorem is proved for a common fixed point of a family of nonexpansive mappings provided that , , satisfy some mild conditions.

Highlights

Let K be a nonempty closed convex subset of a real Banach space E
A mapping T : K → E is called nonexpansive if Tx − T y ≤ x − y for all x, y ∈ K
For a sequence {αn} of real numbers in (0, 1) and an arbitrary u ∈ K, let the sequence {xn} in K be iteratively defined by x0 ∈ K, xn+1 := αn+1u + 1 − αn+1 Txn, n ≥ 0

Summary

Introduction

Let K be a nonempty closed convex subset of a real Banach space E. Shioji and Takahashi [10] extended Wittmann’s result to Banach spaces with uniformly Gateaux differentiable norms and in which each nonempty closed convex subset of K has the fixed point property for nonexpansive mappings and {αn} satisfies conditions (i), (ii), and (iii)∗. R, is a proper subset of E and Ti maps K into E, the iteration process (1.4) may fail to be well defined (see (1.3)) It is our purpose in this paper to define an algorithm for nonself-mappings and to obtain a strong convergence theorem to a fixed point of a family of nonself nonexpansive mappings in Banach spaces more general than the spaces considered by Takahashi et al [11] with {αn} satisfying conditions (i), (ii), and (iii)∗. To more general Banach spaces and to the class of nonself -maps

Preliminaries

Main results